Modular forms of negative weight

In 1939 Rademacher derived a conditionally convergent series
expression for the elliptic modular invariant. This motivated
investigations by various authors in to the problem of constructing
modular functions, and even modular forms of negative weight, for
discrete groups of isometries of the hyperbolic plane.

We will describe a generalization of Rademacher’s construction that
furnishes spanning sets for a certain subspace of the space of
meromorphic modular forms of even integral weight, for any group
commensurable with the modular group. In the course of this we are
led to an analytic continuation of the elliptic modular invariant,
and an association of Dirichlet series to groups commensurable with
the modular group.

The expressions we obtain behave well under the actions of Hecke
operators and have simple branching rules. These facts lead to
applications in monstrous moonshine and three dimensional quantum
gravity.